Roman numerals explained

The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols:[1]

Symbol

Value

1

5

10

50

100

500

1,000

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Hindu-Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some minor applications to this day.

Roman numeric system

Basic decimal pattern

The original pattern for Roman numerals used the symbols I, V. and X (1, 5, and 10) as simple tally marks. Each marker for 1 added a unit value up to 5, and was then added to to make the numbers from 6 to 9:

.The numerals for 4 and 9 proved problematic (among other things, they are easily confused with and), and are generally replaced with (one less than 5) and (one less than 10). This feature of Roman numerals is called subtractive notation.

The numbers from 1 to 10 (including subtractive notation for 4 and 9) are expressed in Roman numerals as follows:

.[2] The system being basically decimal, tens and hundreds follow the same pattern:

Thus 10 to 100 (counting in tens, with taking the place of, taking the place of and taking the place of):

.Note that 40 and 90 follow the same subtractive pattern as 4 and 9.

Similarly, 100 to 1000 (counting in hundreds):

.

Many numbers include hundreds, units and tens. The Roman numeral system being basically decimal, each "place" is added separately, in descending sequence from left to right, as with "arabic" numbers. For example, the number 39 is , (three tens and a ten less one), 246 is (two hundreds, a fifty less ten, a five and a one. As each place has its own notation there is no need for place keeping zeros, so "missing places" can be simply omitted: thus 207, for instance, is written (two hundreds, a five and two ones) and 1066 becomes (a thousand, a fifty and a ten, a five and a one)[3][4]

Roman numerals for large numbers are nowadays seen mainly in the form of year numbers (other uses are detailed later in this article), as in these examples:

1776 as, the date written on the book held by the Statue of Liberty.[5]

Alternative forms

The "standard" forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval and modern times.[7]

Inscriptions dating from the Roman period not infrequently use "additive" forms such as and for "4" and "9" instead of and . There are even instances of both forms appearing within the same document.

While in standard usage subtractive notation is limited to use of "adjacent" numerals, so that 99 (for example) is written, forms such as are not unknown.

or are sometimes used for "18" instead of . The Latin word for "eighteen" is often rendered as the equivalent of "twenty less two", which may be the source of this usage.

Sometimes and are not used, with instances such as and rather than or .[8][9]

At the beginning of the 20th century, different representations of 900 (conventionally) appeared in several inscribed dates. For instance, 1910 is shown on Admiralty Arch, London, as rather than, while on the north entrance to the Saint Louis Art Museum, 1903 is inscribed as rather than .[13]

History

Pre-Roman times and ancient Rome

Although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used,,,,, and for,,,,, and, of which only and happened to be letters in their alphabet.

Hypotheses about the origin of Roman numerals

Tally marks

One hypothesis is that the Etrusco-Roman numerals actually derive from notches on tally sticks, which continued to be used by Italian and Dalmatian shepherds into the 19th century.[14]

Thus, descends not from the letter but from a notch scored across the stick. Every fifth notch was double cut i.e.,,,, etc.), and every tenth was cross cut,), much like European tally marks today. This produced a positional system: Eight on a counting stick was eight tallies,, or the eighth of a longer series of tallies; either way, it could be abbreviated (or), as the existence of a implies four prior notches. By extension, eighteen was the eighth tally after the first ten, which could be abbreviated, and so was . Likewise, number four on the stick was the I-notch that could be felt just before the cut of the, so it could be written as either or . Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were transferred to writing, the marks were easily identified with the existing Roman letters, and .

The tenth or along the stick received an extra stroke. Thus 50 was written variously as,,,,, etc., but perhaps most often as a chicken-track shape like a superimposed and : . This had flattened to (an inverted T) by the time of Augustus, and soon thereafter became identified with the graphically similar letter . Likewise, 100 was variously,,,, or as any of the symbols for 50 above plus an extra stroke. The form (that is, a superimposed and like:) came to predominate. It was written variously as or, was then abbreviated to or, with variant finally winning out because, as a letter, it stood for Latin: ''centum'', Latin for "hundred".

The hundredth or was marked with a box or circle. Thus 500 was like a superimposed on a or, becoming or by the time of Augustus, under the graphic influence of the letter . It was later identified as the letter D; an alternative symbol for "thousand" was (or or), and half of a thousand or "five hundred" is the right half of the symbol, (or or), and this may have been converted into .[15] This at least was the etymology given to it later on.

Meanwhile, 1000 was a circled or boxed :,,, and by Augustinian times was partially identified with the Greek letter phi. Over time, the symbol changed to and . The latter symbol further evolved into, then, and eventually changed to under the influence of the Latin word Latin: ''mille'' "thousand".

Hand signals

Alfred Hooper has an alternative hypothesis for the origin of the Roman numeral system, for small numbers.[16] Hooper contends that the digits are related to hand gestures for counting. For example, the numbers,,, correspond to the number of fingers held up for another to see., then represents that hand upright with fingers together and thumb apart. Numbers 6–10, are represented with two hands as follows (left hand, right hand) 6=, 7=, 8=, 9=, 10= and results from either crossing of the thumbs, or holding both hands up in a cross.

Another possibility is that each represents a finger and represents the thumb of one hand. This way the numbers between 1–10 can be counted on one hand using the order: =P, =PR, =PRM, =IT, =T, =TP, =TPR, =TPRM, =IN, =N (P=Pinky, R=Ring, M=Middle, I=Index, T=Thumb N=No Fingers/Other Hand). This pattern can also be continued using the other hand with the fingers representing and the thumb .

Intermediate symbols deriving from few original symbols

A third hypothesis about the origins states that the basic ciphers were,, and (or) and that the intermediary ones were derived from taking half of those (half an is, half a is and half a is).[17] The was later replaced with, the initial of Mille (the Latin word for "thousand").

Middle Ages and Renaissance

Minuscule (lower-case) letters were developed in the Middle Ages, well after the demise of the Western Roman Empire, and since that time lower-case versions of Roman numbers have also been commonly used:,,,, and so on.

Numerals in documents and inscriptions from the Middle Ages sometimes include additional symbols, which today are called "medieval Roman numerals". Some simply substitute another letter for the standard one (such as "" for "", or "" for ""), while others serve as abbreviations for compound numerals ("" for "", or "" for ""). Although they are still listed today in some dictionaries, they are long out of use.[20]

Could also stand for 2 (see also, the symbol for the dupondius). From a barring of two I's.

250

300

400

,

500

Redundant with, abbreviates Latin: quingenti, Latin for 500.

2000

Chronograms, messages with dates encoded into them, were popular during the Renaissance era. The chronogram would be a phrase containing the letters,,,,,, and . By putting these letters together, the reader would obtain a number, usually indicating a particular year.

Modern use

By the 11th century, Hindu–Arabic numerals had been introduced into Europe from al-Andalus, by way of Arab traders and arithmetic treatises. Roman numerals, however, proved very persistent, remaining in common use in the West well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Replacement by their more convenient "Arabic" equivalents was quite gradual, and Roman numerals are still used today in certain contexts. A few examples of their current use are:

Generational suffixes, particularly in the US, for people sharing the same name across generations, for example William Howard Taft IV.

In the French Republican Calendar, initiated during the French Revolution, years were numbered by Roman numerals – from the year I (1792) when this calendar was introduced to the year XIV (1805) when it was abandoned.

The year of production of films, television shows and other works of art within the work itself. It has been suggested – by BBC News, perhaps facetiously – that this was originally done "in an attempt to disguise the age of films or television programmes."[23] Outside reference to the work will use regular Hindu–Arabic numerals.

In computing, Roman numerals may be used in identifiers which are limited to alphabetic characters by syntactic constraints of the programming language. In LaTeX, for instance, \labelitemiii refers to the label of an item in the third level of a nested list environment.

In military unit designation, Roman numerals are often used to distinguish between units at different levels. This reduces possible confusion, especially when viewing operational or strategic level maps. In particular, army corps are often numbered using Roman numerals (for example the American XVIII Airborne Corps or the WW2-era German III Panzerkorps) with Hindu-Arabic numerals being used for divisions and armies.

In entomology, the broods of the thirteen and seventeen year periodical cicadas are identified by Roman numerals.

In advanced mathematics (including trigonometry, statistics, and calculus), when a graph includes negative numbers, its quadrants are named using,,, and . These quadrant names signify positive numbers on both axes, negative numbers on the X axis, negative numbers on both axes, and negative numbers on the Y axis, respectively. The use of Roman numerals to designate quadrants avoids confusion, since Hindu-Arabic numerals are used for the actual data represented in the graph.

Modern use in continental Europe

Some uses that are rare or never seen in English speaking countries are relatively common in continental Europe. For instance:

Capital or small capital Roman numerals are widely used in Romance languages to denote, e.g. the French French: {{Smallcaps|xviiie siècle}}[26] and the Spanish Spanish; Castilian: siglo XVIII mean "18th century". Slavic languages in and adjacent to Russia similarly favour Roman numerals (Russian: XVIII век). On the other hand, in Slavic languages in Central Europe, like most Germanic languages, one writes "18." (with a period) before the local word for "century".

Mixed Roman and Hindu-Arabic numerals are sometimes used in numeric representations of dates (especially in formal letters and official documents, but also on tombstones). The is written in Roman numerals, while the day is in Hindu-Arabic numerals: "14..1789" and ".14.1789" both refer unambiguously to 14 June 1789.

Roman numerals are sometimes used to represent the in hours-of-operation signs displayed in windows or on doors of businesses,[27] and also sometimes in railway and bus timetables. Monday, taken as the first day of the week, is represented by . Sunday is represented by .The hours of operation signs are tables composed of two columns where the left column is the day of the week in Roman numerals and the right column is a range of hours of operation from starting time to closing time. In the example case (left), the business opens from 10 AM to 7 PM on weekdays, 10 AM to 5 PM on Saturdays and is closed on Sundays. Note that the listing uses 24-hour time.

Roman numerals may also be used for floor numbering.[28][29] For instance, apartments in central Amsterdam are indicated as 138-, with both a Hindu-Arabic numeral (number of the block or house) and a Roman numeral (floor number). The apartment on the ground floor is indicated as 'Dutch; Flemish: 138-huis'.

In Italy, where roads outside built-up areas have kilometre signs, major roads and motorways also mark 100-metre subdivisionals, using Roman numerals from to for the smaller intervals. The sign " | 17" thus marks kilometre 17.9.

A notable exception to the use of Roman numerals in Europe is in Greece, where Greek numerals (based on the Greek alphabet) are generally used in contexts where Roman numerals would be used elsewhere.

Special values

Zero

The number zero does not have its own Roman numeral, but the word nulla (the Latin word meaning "none") was used by medieval scholars in lieu of 0. Dionysius Exiguus was known to use nulla alongside Roman numerals in 525.[30][31] About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for "nothing"), in a table of epacts, all written in Roman numerals.[32]

Fractions

Though the Romans used a decimal system for whole numbers, reflecting how they counted in Latin, they used a duodecimal system for fractions, because the divisibility of twelve makes it easier to handle the common fractions of 1/3 and 1/4 than does a system based on ten On coins, many of which had values that were duodecimal fractions of the unit Latin: ''[[As (coin)|as]]'', they used a tally-like notational system based on twelfths and halves. A dot (•) indicated an Latin: ''uncia'' "twelfth", the source of the English words inch and ounce; dots were repeated for fractions up to five twelfths. Six twelfths (one half) was abbreviated as the letter S for Latin: ''semis'' "half". Uncia dots were added to S for fractions from seven to eleven twelfths, just as tallies were added to for whole numbers from six to nine.[33]

Each fraction from 1/12 to 12/12 had a name in Roman times; these corresponded to the names of the related coins:

The arrangement of the dots was variable and not necessarily linear. Five dots arranged like (⁙) (as on the face of a die) are known as a quincunx, from the name of the Roman fraction/coin. The Latin words Latin: sextans and Latin: quadrans are the source of the English words sextant and quadrant.

Other Roman fractional notations included the following:

1/8 Latin: ''sescuncia, sescunciae'' (from sesqui- + uncia, i.e. 1½ uncias), represented by a sequence of the symbols for the semuncia and the uncia.

Large numbers

A number of systems were developed for the expression of larger numbers that cannot be conveniently expressed using the normal seven letter symbols of conventional Roman numerals.

Apostrophus

One of these was the apostrophus,[34] in which 500 (usually written as "") was written as |, while 1,000 was written as | instead of "".[15] This is a system of encasing numbers to denote thousands (the s and s functioned in this case as the Roman equivalent of parentheses), and has its origins in Etruscan numeral usage. The and used to represent 500 and 1,000 in conventional Roman numerals were probably derived from | and |, respectively.

In this system, an extra | denoted 500, | 5,000 and | 50,000. For example:

Base number

= 1,000

= 10,000

= 100,000

with |

| = 500

= 1,500

= 10,500

= 100,500

with |

| = 5,000

= 15,000

= 105,000

with |

| = 50,000

= 150,000

Sometimes | was reduced to ↀ for 1,000. John Wallis is often credited for introducing the symbol for infinity (modern ∞), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, | for 5,000 was reduced to ↁ; | for 10,000 to ↂ; | for 50,000 to ↇ; and | for 100,000 to ↈ.[14]

Vinculum

Another system is the vinculum, where a conventional Roman numeral is multiplied by 1,000 by adding an overline.[14] Although mathematical historian David Eugene Smith disputes that this was part of ancient Roman usage, the notation was certainly in use in the Middle Ages, and is at least a convenient and "conventional" way to express numbers greater than 3,999 in Roman numerals.

For instance:

for 4,000

for 25,000

Adding vertical lines (or brackets) before and after the numeral seems to have been used (at least by late medieval times) to multiply a Roman numeral by 10: thus || for 10,000 rather than (perhaps under the influence of Arabic/Hindu numerals, which were by this time coming into use). In combination with the overline the bracketed forms might be used to raise the multiplier to (say) ten (or one hundred) thousand, thus:

|| for 80,000 (or 800,000)

|| for 200,000 (or 2,000,000)

This needs to be distinguished from the custom of adding both underline and overline to a Roman numeral, simply to make it clear that it is a number, e.g. .

Notes and References

Book: Gordon, Arthur E.. Illustrated Introduction to Latin Epigraphy. University of California Press. 1982. 0520050797. Berkeley. Alphabetic symbols for larger numbers, such as for 500,000, have also been used to various degrees of standardization..

Web site: Gallery: Museum's North Entrance (1910). Saint Louis Art Museum. 10 January 2014. The inscription over the North Entrance to the Museum reads: "Dedicated to Art and Free to All · ". These Roman numerals translate to 1903, indicating that the engraving was part of the original building designed for the 1904 World's Fair..

Book: Georges. Ifrah. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E. F. Harding, Sophie Wood, Ian Monk. John Wiley & Sons. 2000.